Perverse Sheaves and Applications to Representation Theory

Description

Since its inception around 1980, the theory of perverse sheaves has been a vital tool of fundamental importance in geometric representation theory. This book, which aims to make this theory accessible to students and researchers, is divided into two parts. The first six chapters give a comprehensive account of constructible and perverse sheaves on complex algebraic varieties, including such topics as Artin's vanishing theorem, smooth descent, and the nearby cycles functor. This part of the book also has a chapter on the equivariant derived category, and brief surveys of side topics including etale and $\ell$-adic sheaves, $\mathcal{D}$-modules, and algebraic stacks. The last four chapters of the book show how to put this machinery to work in the context of selected topics in geometric representation theory: Kazhdan-Lusztig theory; Springer theory; the geometric Satake equivalence; and canonical bases for quantum groups. Recent developments such as the $p$-canonical basis are also discussed. The book has more than 250 exercises, many of which focus on explicit calculations with concrete examples. It also features a 4-page ``Quick Reference'' that summarizes the most commonly used facts for computations, similar to a table of integrals in a calculus textbook. Preface Chapter 1. Sheaf theory 1.1. Sheaves 1.2. Pullback, push-forward, and base change 1.3. Open and closed embeddings 1.4. Tensor product and sheaf Hom 1.5. The right adjoint to proper push-forward 1.6. Relations among natural transformations 1.7. Local systems 1.8. Homotopy 1.9. More base change theorems 1.10. Additional notes and exercises Chapter 2. Constructible sheaves on complex algebraic varieties 2.1. Preliminaries from complex algebraic geometry 2.2. Smooth pullback and smooth base change 2.3. Stratifications and constructible sheaves 2.4. Divisors with simple normal crossings 2.5. Base change and the affine line 2.6. Artin’s vanishing theorem 2.7. Sheaf functors and constructibility 2.8. Verdier duality 2.9. More compatibilities of functors 2.10. Localization with respect to a \Gm-action 2.11. Homology and fundamental classes 2.12. Additional notes and exercises Chapter 3. Perverse sheaves 3.1. The perverse 𝑡-structure 3.2. Tensor product and sheaf Hom for perverse sheaves 3.3. Intersection cohomology complexes 3.4. The noetherian property for perverse sheaves 3.5. Affine open subsets and affine morphisms 3.6. Smooth pullback 3.7. Smooth descent 3.8. Semismall maps 3.9. The decomposition theorem and the hard Lefschetz theorem 3.10. Additional notes and exercises Chapter 4. Nearby and vanishing cycles 4.1. Definitions and preliminaries 4.2. Properties of algebraic nearby cycles 4.3. Extension across a hypersurface 4.4. Unipotent nearby cycles 4.5. Beilinson’s theorem 4.6. Additional notes and exercises Chapter 5. Mixed sheaves 5.1. Étale and ℓ-adic sheaves 5.2. Local systems and the étale fundamental group 5.3. Passage to the algebraic closure 5.4. Mixed ℓ-adic sheaves 5.5. \scD-modules and the Riemann–Hilbert correspondence 5.6. Mixed Hodge modules 5.7. Further topics around purity Chapter 6. Equivariant derived categories 6.1. Preliminaries on algebraic groups, actions, and quotients 6.2. Equivariant sheaves and perverse sheaves 6.3. Twisted equivariance 6.4. Equivariant derived categories 6.5. Equivariant sheaf functors 6.6. Averaging, invariants, and applications 6.7. Equivariant cohomology 6.8. The language of stacks 6.9. Fourier–Laumon transform 6.10. Additional exercises Chapter 7. Kazhdan–Lusztig theory 7.1. Flag varieties and Hecke algebras 7.2. Convolution 7.3. The categorification theorem 7.4. Mixed sheaves on the flag variety 7.5. Parity sheaves 7.6. Soergel bimodules 7.7. Additional exercises Chapter 8. Springer theory 8.1. Nilpotent orbits and the Springer resolution 8.2. The Springer sheaf 8.3. The Springer correspondence 8.4. Parabolic induction and restriction 8.5. The generalized Springer correspondence 8.6. Additional exercises Chapter 9. The geometric Satake equivalence 9.1. The affine flag variety and the affine Grassmannian 9.2. Convolution 9.3. Categorification of the affine and spherical Hecke algebras 9.4. The Satake isomorphism 9.5. Exactness and commutativity 9.6. Weight functors 9.7. Standard sheaves and Mirković–Vilonen cycles 9.8. Hypercohomology as a fiber functor 9.9. The geometric Satake equivalence 9.10. Additional exercises Chapter 10. Quiver representations and quantum groups 10.1. Quiver representations 10.2. Hall algebras and quantum groups 10.3. Convolution 10.4. Canonical bases for quantum groups 10.5. Mixed Hodge modules and categorification 10.6. Mixed ℓ-adic sheaves and the Hall algebra 10.7. Additional exercises Appendix A. Category theory and homological algebra A.1. Categories and functors A.2. Monoidal categories A.3. Additive and abelian categories A.4. Triangulated categories A.5. Chain complexes and the derived category A.6. Derived functors A.7. 𝑡-structures A.8. Karoubian and Krull–Schmidt categories A.9. Grothendieck groups A.10. Duality for rings of finite global dimension Appendix B. Calculations on \Cⁿ B.1. Holomorphic maps and cohomology B.2. Constructible sheaves on \Cⁿ, I B.3. Local systems on open subsets of \C B.4. Constructible sheaves on \Cⁿ, II B.5. Nearby cycles on \Cⁿ B.6. Equivariant sheaves on \C Quick reference QR\arabic{section}.Triangulated categories QR\arabic{section}.Sheaves QR\arabic{section}.Constructible sheaves QR\arabic{section}.Perverse sheaves QR\arabic{section}.Mixed sheaves QR\arabic{section}.Equivariant sheaves Bibliography Index of notation Index